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A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map

Dan Burghelea

Algebraic & Geometric Topology 18 (2018) 3037–3087

For f : X S1 a continuous angle-valued map defined on a compact ANR X, κ a field and any integer r 0, one proposes a refinement δrf of the Novikov–Betti numbers of the pair (X,ξf) and a refinement δ̂rf of the Novikov homology of (X,ξf), where ξf denotes the integral degree one cohomology class represented by f. The refinement δrf is a configuration of points, with multiplicity located in 2 identified to 0, whose total cardinality is the r th Novikov–Betti number of the pair. The refinement δ̂rf is a configuration of submodules of the r th Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of δrf. When κ = , the configuration δ̂rf is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L2–homology of the infinite cyclic cover of X defined by f, which is an L(S1)–Hilbert module. One discusses the properties of these configurations, namely robustness with respect to continuous perturbation of the angle-values map and the Poincaré duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov–Betti numbers replacing standard homology and standard Betti numbers.

Novikov–Betti numbers, angle-valued maps, barcodes
Mathematical Subject Classification 2010
Primary: 46M20, 55N35, 57R19
Received: 21 November 2017
Revised: 12 March 2018
Accepted: 23 March 2018
Published: 22 August 2018
Dan Burghelea
Department of Mathematics
The Ohio State University
Columbus, OH
United States